Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
10.2 Variational regularization of the Cauchy problem
T V(f)= ∇f1().
This functional, while very natural, generates a nonlinear Euler equation for a minimum point, replacing, for example, αf(;α) in the equation (10.1.4) with
−αdiv(|∇f|−1∇f). The computational effort is accordingly greater, but the qual- ity of the reconstruction of the discontinuous f improves. For an analysis of the numerical solution when regularizing by the total variation functional, we refer to the paper of Dobson and Santosa [DoS]. A particular and very effective method of recovery of discontinuuty surface of conductivity based on single layer repre- sentation is given by Kwon, Seo, and Yoon [KwSY].
10.2 Variational regularization of the Cauchy problem
The numerical method of variational regularization can be considered as a con- cretization of regularization of differential problems. It was developed recently by Klibanov and his collaborators, and we describe it using the results of the paper by Klibanov and Rakesh [KlR], where they discussed the casea0=1 andγ =∂.
We consider the hyperbolic equation
(10.2.1) (a02∂t2u+Au)= f inQ=×(−T,T),u∈ H(2)(Q), with Cauchy data
(10.2.2) u=g0, ∂νu=g1 on=γ×(−T,T).
We assume thata0=a0(x)∈C1() and is positive onand thatAis the elliptic operator−+c,c∈ L∞(Q). Generally, this problem is not well-posed, and even when one has Lipschitz stability (as in Theorem 3.4.5), it is overdetermined, so its numerical solution needs some version of the least-squares method. When solving the Cauchy problem (10.2.1)–(10.2.2), we can assume thatg0=0,g1 =0. Indeed, we can always achieve this by extending the Cauchy data ontoas a functionu∗ and subtractingu∗fromu, obtaining Cauchy data for the difference zero. The ex- tension operator is continuous from H(3/2)(γ×(−T,T))×H(1/2)(γ×(−T,T)) and can be explicitly written down for simple domains. Later on, we assume thatg0=g1=0.
The general regularization scheme (2.3.3) is quite applicable here. Specifically, we can solve the following (regularized) minimization problem:
(10.2.3) min(a20∂t2v+Av− f22(Q)+αv2•(2)(Q)),v∈H˚(2)(Q;),
wherev•(2)(Q) is one of the equivalent norms inH(2)(Q) defined as
1≤j≤n+1
∂2jv22(Q)+ v22(Q) 1/2
and ˚H(2)(Q;) is the subspace ofH(2)(Q) formed of functionsvwithv=∂νv=0 on. Due to coercivity and convexity of the regularized functional (10.2.3) on the Hilbert space ˚H(2)(Q;), the minimum pointu(;α) in this space exists and is unique by known basic results of convex analysis; see the book of Ekeland and Temam [ET]. To describe the Euler equation for the minimum pointu(;α) we need a new scalar product [,](Q) inH(2)(Q), defined as follows:
(10.2.4) [u,v](Q)=
Q
a02∂t2u+Au
a02∂t2v+Av +α
1≤j≤n+1
∂2ju∂2jv+αuv. Then a minimum point of (10.2.3) is a minimum point of the quadratic functional
[v,v](Q)−2
f,a02∂t2v+Av 2(Q), and by standard technique we have the variational Euler equation (10.2.5) [u(;α),v] (Q)=
f,a20∂t2v+Av 2(Q),v∈H˚(2)(Q;), for the minimum pointu(;α).
Lemma 10.2.1. We have
u−u( ;α)•(2)(Q)≤ u•(2)(Q),
(a02∂t2+A)(u−u( ;α))2(Q)≤2−1/2α1/2u•(2)(Q). (10.2.6)
PROOF. Sinceusolves equation (10.2.1), from the definition (10.2.4) of the scalar product [,] we have
[u,v](Q)= f,
a02∂t2v+Av
2(Q)+α(u,v)•(2)(Q).
Subtracting this equality from the definition (10.2.5) of the weak solutionu( ;α), we obtain
[u( ;α)−u,v](Q)= −α(u,v)•2(Q), v∈H˚(2)(Q;).
Lettingv=u( ;α)−uand using the definition (10.2.4) of the scalar product [,] gives
(a02∂t2+A)(u( ;α)−u)22(Q)+αu( ;α)−u2•(2)(Q)
= −α(u,u( ;α)−u)•(2)(Q)≤αu•(2)(Q)u( ;α)−u•(2)(Q)
≤α/2u( ;α)−u2•(2)(Q)+α/2u2•(2)(Q), (10.2.7)
where we used the Schwarz inequality for the scalar product (,)•(2)and the elemen- tary inequalityab≤ 12(a2+b2). Subtracting the first term of the last quantity in
10.2. Variational regularization of the Cauchy problem 305 (10.2.7) and dropping the first term of the first quantity we obtain the first inequality (10.2.6). Dropping the second term gives the second inequality (10.2.6).
The proof is complete.
Lemma 10.2.1 does not guarantee thatu( ;α) is convergent touinL2(Q) when α→0 (and generally this is false, due to nonuniqueness in the Cauchy problem inQ). However, combining the results of Lemma 10.2.1 and of Theorems 3.4.1 and 3.4.5, one can obtain convergence in domains Qε determined by a weight function in Carleman estimates for the domain Q=×(−T,T) in the cases (3.4.2), (3.4.3), or even in all ofQunder the conditions of Theorem 3.4.5.
Corollary 10.2.2. If the domain Q and the coefficient a0satisfy the conditions of Theorem 3.4.1, then
(10.2.8) u( ;α)−u(1)(Qε)≤Cαλu(2)(Q), whereλ∈(0,1)depends onε >0determining the domain Qε.
If the domain Q and the coefficient a0satisfy the conditions of Theorem 3.4.5 andγ =∂then
(10.2.9) u(;α)−u(1)(Q)≤Cα1/2u(2)(Q).
PROOF. To prove (10.2.8) we observe that the first bound (10.2.6) implies that u( ;α)−u(2)(Q)≤Cu(2)(Q), and we combine the second bound (10.2.6) and the bound (3.4.7) of Theorem 3.4.1.
To prove (10.2.9) we observe that Theorem 3.4.5 implies the bound w(1)(Q)≤C(a02∂t2+A)w2(Q),
provided thatw ∈H(2)(Q) andw=∂νw =0 on∂×(−T,T). Letw=u( ;α)−
uand use the second bound (10.2.6).
In the paper [K1R] there are numerical results for this regularization in the case of the wave equation inR2. The authors found solutionsu( ;α) of the regu- larized problem (after discretization) by a finite-element method. The numerical experiment agrees with the convergence estimate very well.
Similarly, one can consider the lateral Cauchy problem for parabolic equations (10.2.10) a0∂tu+Au= f inQ, u∈ H2,1;2(Q),
with the lateral boundary data
(10.2.11) u=g0, ∂νu =g1 on×(−T,T).
In this case we keep the assumptions abouta0while considering a general elliptic operator A of second order with time-dependent coefficients, C1()-principal coefficients, and other coefficients in L∞(Q). As above, for a solution of this problem we can assume thatg0=g1=0.
In the case of parabolic equations, we define v2•2,1;2(Q)= ∂tv22(Q)+
1≤j≤n
∂2jv22(Q)+ v22(Q).
Exercise 10.2.3. Show that there is a unique minimizeru( ;α) of the problem min(a02∂tv+Av22(Q)+αv2•2,1;2(Q))
overv∈ H2,1;2(Q) withv=∂νv=0 onγ×(−T,T). Show that u−u( ;α)•2,1;2(Q)≤ u•2,1;2(Q),
(a0∂tv+Av)(u−u( ;α)2(Q)≤2−1/2α1/2u•2,1;2(Q) (10.2.12)
and derive from Theorem 3.3.10 that for any domainQεwith closure inQ∪(γ× (−T,T)) there areCandλ∈(0,1) such that
u−u( ;α)2,1;2(Qε)≤Cαλu2,1;2(Q).
Also, one can solve in this manner the backward initial problem for equation (10.2.10) with final and lateral boundary data
(10.2.13) u(,T)=uT, on× {T},u=0 on∂×(0,T).
As above, we can use extension results and consideruT =0.
Exercise 10.2.4. Show that there is a unique solutionu( ;α) of the minimization problem
min(∂tv+Av− f22(Q)+αv2•2,1;2(Q)) overv∈ H2,1;2(Q) withv=0 on∂×(0,T) and on× {0}.
Further, show that
(u−u( ;α))(t)2()≤C(αλ+α1/2)u2,1;2(Q), λ=λ(t)∈(0,1), and that forA= −+cone can chooseλ(t)=t/(2T).
{Hint: Again make use of the scheme of the proof of Lemma 10.2.1 to obtain the bounds (10.1.12). Then the differencew =u−u( ;α) will solve the parabolic equation (10.2.10) with the right side f1such that
f12(Q)≤Cα1/2u2,1;2(Q)
and will have zero boundary and final data. Use the solutionw1 of the initial boundary value problem for (10.2.10) with right side f1and with zero boundary and initial data and bound this solution using Theorem 9.1 and the trace theorem.
to bound the solution of the homogeneous parabolic equation for w−w1 use Theorem 3.1.3 and Example 3.1.6.}
So far, we have considered only linear problems or linearizations, while identi- fication of coefficients creates at least quadratic nonlinearity. Some Newton-type
10.2. Variational regularization of the Cauchy problem 307 algorithms make use of similar regularization and have been proved to be effective, but there is no general global approach to this problem.
Another interesting “alternating” iterative method has been suggested in appli- cations and considered and justified in general framework in the paper of Kozlov and Maz’ya [KozM]. For the Laplace equation in a domainwith Cauchy data given on ⊂∂it consists in first solving the mixed boundary value problem with given data onand zero Neumann data on∂\; then solving the mixed boundary value problem with given Neumann data onand the Dirichlet data on the remaining part, which are obtained at the previous step; and then iterating.
This algorithm is convergent, and it would be interesting to develop its analogue for the problems of identification of coefficients.
As an example of efficient numerical solution of practically important problem we consider regularization of a linear integral equation arising in nearfield acousti- cal holography. This technique seeks for vibrations of a surface from the acoustical pressure generated by these vibrations. We will describe recent results of DeLillo, Isakov, Valdivia, and Wang who handled a problem from aircraft industry. The componentuof pressure of frequencyksatisfies the Helmholtz equation
u+k2u =0 in.
Hereis a Lipschitz domain inR3with connectedR3\. Microphones are located on a surface0inside (cabin), so we are given
u=g0on 0. One is looking for the so-called normal velocity
v=∂νuon=∂.
It was shown in [DIVW] that anyu ∈H(1)() admits unique representation by the single layer potential distributed over(disregard of Dirichlet or Neumann eigenvalues of the Laplacian in). So it was proposed in [DIVW] to solve for densityφfrom the integral equation
(10.2.14) 1/(4π)
ei k|x−y|/|x−y|φ(y)d(y)=u(x), x∈0
and then to find∂νu on from known jump relations for normal derivative of single layer potential. Uniqueness can be guaranteed if 0 is a boundary of a domain of small volume. Under constraints similar to those in Theorem 3.3.1 one has logarithmic stability, but since distance from0 tois relatively small one can achieve high resolution.
The integral equation (10.2.14) was discretized in [DIVW] by using piecewise linear triangular boundary elements and the resulting linear algebraic system with Nunknowns was solved by the iterative conjugate gradient method where number Jof iterations plays the role of regularization parameter. The choice ofJis cricial for efficient numerics. Whenis a cylinder of radius 1 with floor and flat endcaps modeling Cessna 650 fuselage,k=3 (typical acoustic range), andN =1000, we used J=30 as suggested by generalized cross-validation. Relative L∞-error of
0.01 in data produced relative reconstruction error of 0.1. Our experience showed that use of integral equations is the most effective in numerical solution of ill-posed Cauchy problems of relatively large size. Of course, it presumes a simple analytic form of a fundamental solution. So far single layer method is the most competitive algorithm for nearfield acoustic holography which is suitable for any geometry of and for exterior problems typical for automotive industry and naval applications.